You are here
HomePspace
Primary tabs
Pspace
Suppose $X$ is a completely regular topological space. Then $X$ is said to be a Pspace if every prime ideal in $C(X)$, the ring of continuous functions on $X$, is maximal.
For example, every space with the discrete topology is a Pspace.
Algebraically, a commutative reduced ring $R$ with $1$ such that every prime ideal is maximal is equivalent to any of the following statements:

$R$ is vonNeumann regular,

every ideal in $R$ is the intersection of prime ideals,

every ideal in $R$ is the intersection of maximal ideals,

every principal ideal is generated by an idempotent.
When $R=C(X)$, then $R$ is commutative reduced with $1$. In addition to the algebraic characterizations of $R$ above, $X$ being a Pspace is equivalent to any of the following statements:

if $f,g\in C(X)$, then $(f,g)=(f^{2}+g^{2})$.
Some properties of Pspaces:
1. Every subspace of a Pspace is a Pspace,
2. Every quotient space of a Pspace is a Pspace,
3. 4. Every Pspace has a base of clopen sets.
For more properties of Pspaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see here.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Mathematics Subject Classification
54E18 no label found16S60 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: numerical method (implicit) for nonlinear pde by roozbe
new question: Harshad Number by pspss
Sep 14
new problem: Geometry by parag
Aug 24
new question: Scheduling Algorithm by ncovella
new question: Scheduling Algorithm by ncovella